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COLLECTIVE RISK MODELS WITH DEPENDENCE UNCERTAINTY

Published online by Cambridge University Press:  03 April 2017

Haiyan Liu  and
Ruodu Wang
Haiyan Liu
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada E-Mail: [email protected]
Ruodu Wang*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada
*
E-Mail: [email protected]
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Abstract

We bring the recently developed framework of dependence uncertainty into collective risk models, one of the most classic models in actuarial science. We study the worst-case values of the Value-at-Risk (VaR) and the Expected Shortfall (ES) of the aggregate loss in collective risk models, under two settings of dependence uncertainty: (i) the counting random variable (claim frequency) and the individual losses (claim sizes) are independent, and the dependence of the individual losses is unknown; (ii) the dependence of the counting random variable and the individual losses is unknown. Analytical results for the worst-case values of ES are obtained. For the loss from a large portfolio of insurance policies, an asymptotic equivalence of VaR and ES is established. Our results can be used to provide approximations for VaR and ES in collective risk models with unknown dependence. Approximation errors are obtained in both cases.

Keywords

Collective risk modelValue-at-RiskExpected Shortfalldependence uncertaintyasymptotic equivalence
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 2 , May 2017 , pp. 361 - 389
Copyright
Copyright © Astin Bulletin 2017 

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References

Albrecher, H., Boxma, O.J. and Ivanovs, J. (2014) On simple ruin expressions in dependent Sparre Andersen risk models. Journal of Applied Probability, 51, 293–296.Google Scholar
Bernard, C., Denuit, M. and Vanduffel, S. (2017a) Measuring portfolio risk under partial dependence information. Journal of Risk and Insurance. DOI: 10.1111/jori.12165.Google Scholar
Bernard, C., Jiang, X. and Wang, R. (2014) Risk aggregation with dependence uncertainty. Insurance: Mathematics and Economics, 54, 93–108.Google Scholar
Bernard, C., Rüschendorf, L. and Vanduffel, S. (2017b) VaR bounds with variance constraint. Journal of Risk and Insurance. DOI: 10.1111/jori.12108.Google Scholar
Bernard, C., Rüschendorf, L., Vanduffel, S. and Wang, R. (2017c) Risk bounds for factor models. Finance and Stochastics, forthcoming.Google Scholar
Bernard, C. and Vanduffel, S. (2015) A new approach to assessing model risk in high dimensions. Journal of Banking and Finance, 58, 166–178.Google Scholar
Bignozzi, V., Puccetti, G. and Rüschendorf, L. (2015) Reducing model risk via positive and negative dependence assumptions. Insurance: Mathematics and Economics, 61 (1), 17–26.Google Scholar
Cai, J., Liu, H. and Wang, R. (2017) Asymptotic equivalence of risk measures under dependence uncertainty. Mathematical Finance. DOI: 10.1111/mafi.12140.Google Scholar
Cai, J. and Tan, K.S. (2007) Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. ASTIN Bulletin, 37 (1), 93–112.Google Scholar
Cheung, E.C.K., Landriault, D., Willmot, G.E. and Woo, J.-K. (2010) Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models. Insurance: Mathematics and Economics, 46 (1), 117–126.Google Scholar
Cheung, K.C., Dhaene, J., Lo, A. and Tang, Q. (2014) Reducing risk by merging counter-monotonic risks. Insurance: Mathematics and Economics, 54 (1), 58–65.Google Scholar
Cont, R., Deguest, R. and Scandolo, G. (2010) Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 10 (6), 593–606.Google Scholar
Delbaen, F. (2012) Monetary Utility Functions. Osaka: Osaka University Press.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M.J. and Kaas, R. (2005) Actuarial Theory for Dependent Risks. Chichester, UK: Wiley.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. and Vyncke, D. (2002) The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Mathematics and Economics, 31 (1), 3–33.Google Scholar
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013) Model uncertainty and VaR aggregation. Journal of Banking and Finance, 37 (8), 2750–2764.Google Scholar
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014) An academic response to Basel 3.5. Risks, 2 (1), 25–48.Google Scholar
Embrechts, P., Wang, B. and Wang, R. (2015) Aggregation-robustness and model uncertainty of regulatory risk measures. Finance and Stochastics, 19 (4), 763–790.Google Scholar
Föllmer, H. and Schied, A. (2011) Stochastic Finance: An Introduction in Discrete Time. 3rd Edition, Berlin: Walter de Gruyter.Google Scholar
Hürlimann, W. (2003) Conditional value-at-risk bounds for compound poisson risks and a normal approximation. Journal of Applied Mathematics, 3, 141–153.Google Scholar
Jakobsons, E., Han, X. and Wang, R. (2016) General convex order on risk aggregation. Scandinavian Actuarial Journal, 2016 (8), 713–740.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008) Modern Actuarial Risk Theory: Using R. Springer-Verlag Berlin Heidelberg: Springer Science & Business Media.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2012) Loss Models: from Data to Decisions. 4th Edition, Hoboken, New Jersey: Wiley.Google Scholar
Kousky, C. and Cooke, R.M. (2009) The unholy trinity: Fat tails, tail dependence, and micro-correlations. Resources for the Future Discussion Paper, 09–36.Google Scholar
Landriault, D., Lee, W., Willmot, G.E. and Woo, J.-K. (2014) A note on deficit analysis in dependency models involving Coxian claim amounts. Scandinavian Actuarial Journal, 5, 405–423.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques and Tools. Revised Edition, Princeton, New Jersey: Princeton University Press.Google Scholar
Müller, A. and Stoyan, D. (2002) Comparison Methods for Statistical Models and Risks. England: Wiley.Google Scholar
Puccetti, G. and Rüschendorf, L. (2012) Computation of sharp bounds on the distribution of a function of dependent risks. Journal of Computational and Applied Mathematics, 236 (7), 1833–1840.Google Scholar
Puccetti, G. and Rüschendorf, L. (2014) Asymptotic equivalence of conservative VaR- and ES-based capital charges. Journal of Risk, 16 (3), 3–22.CrossRefGoogle Scholar
Puccetti, G., Rüschendorf, L. and Manko, D. (2016) VaR bounds for joint portfolios with dependence constraints. Dependence Modeling, 4, 368–381.Google Scholar
Puccetti, G., Rüschendorf, L., Small, D. and Vanduffel, S. (2017) Reduction of Value-at-Risk bounds via independence and variance information. Scandinavian Actuarial Journal, 2017 (3), 245–266.Google Scholar
Puccetti, G., Wang, B. and Wang, R. (2013) Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insurance: Mathematics and Economics, 53 (3), 821–828.Google Scholar
Puccetti, G. and Wang, R. (2015) Extremal dependence concepts. Statistical Science, 30 (4), 485–517.Google Scholar
Rüschendorf, L. (2013) Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Heidelberg: Springer.Google Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. Springer Science+Business Media, New York: Springer Series in Statistics.Google Scholar
Svindland, G. (2008) Convex risk measures beyond bounded risk. Ph.D. Thesis, Ludwig Maximilian Universität München.Google Scholar
Tchen, A.H. (1980) Inequalities for distributions with given marginals. Annals of Probability, 8 (4), 814–827.Google Scholar
Wang, B. and Wang, R. (2015) Extreme negative dependence and risk aggregation. Journal of Multivariate Analysis, 136, 12–25.Google Scholar
Wang, R., Bignozzi, V. and Tsakanas, A. (2015) How superadditive can a risk measure be? SIAM Jounral on Financial Mathematics, 6, 776–803.Google Scholar
Wang, R., Peng, L. and Yang, J. (2013) Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. Finance and Stochastics, 17 (2), 395–417.Google Scholar